MinT - Best Known (57, 57+16, s)-Nets in Base 128

Best Known (57, 57+16, s)-Nets in Base 128

(57, 57+16, 1048833)-Net over F₁₂₈ — Constructive and digital

Digital (57, 73, 1048833)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 12, 258)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 8, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
2. digital (45, 61, 1048575)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁶¹, 1048575, F₁₂₈, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(128⁶¹, 8388600, F₁₂₈, 16) (dual of [8388600, 8388539, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁶¹, large, F₁₂₈, 16) (dual of [large, large−61, 17]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9256395Â | 128⁴âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(57, 57+16, 1065087)-Net in Base 128 — Constructive

(57, 73, 1065087)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (12, 20, 16512)-net over F₁₂₈, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 129)-net over F₁₂₈, using
      - s-reduction based on digital (0, 0, s)-net over F₁₂₈ with arbitrarily large s, using
        arbitrary set of 128⁰ points [i]
    2. digital (0, 0, 129)-net over F₁₂₈ (see above)
    3. digital (0, 0, 129)-net over F₁₂₈ (see above)
    4. digital (0, 0, 129)-net over F₁₂₈ (see above)
    5. digital (0, 0, 129)-net over F₁₂₈ (see above)
    6. digital (0, 0, 129)-net over F₁₂₈ (see above)
    7. digital (0, 0, 129)-net over F₁₂₈ (see above)
    8. digital (0, 0, 129)-net over F₁₂₈ (see above)
    9. digital (0, 0, 129)-net over F₁₂₈ (see above)
    10. digital (0, 0, 129)-net over F₁₂₈ (see above)
    11. digital (0, 0, 129)-net over F₁₂₈ (see above)
    12. digital (0, 0, 129)-net over F₁₂₈ (see above)
    13. digital (0, 0, 129)-net over F₁₂₈ (see above)
    14. digital (0, 0, 129)-net over F₁₂₈ (see above)
    15. digital (0, 0, 129)-net over F₁₂₈ (see above)
    16. digital (0, 0, 129)-net over F₁₂₈ (see above)
    17. digital (0, 0, 129)-net over F₁₂₈ (see above)
    18. digital (0, 0, 129)-net over F₁₂₈ (see above)
    19. digital (0, 0, 129)-net over F₁₂₈ (see above)
    20. digital (0, 0, 129)-net over F₁₂₈ (see above)
    21. digital (0, 0, 129)-net over F₁₂₈ (see above)
    22. digital (0, 0, 129)-net over F₁₂₈ (see above)
    23. digital (0, 0, 129)-net over F₁₂₈ (see above)
    24. digital (0, 0, 129)-net over F₁₂₈ (see above)
    25. digital (0, 0, 129)-net over F₁₂₈ (see above)
    26. digital (0, 0, 129)-net over F₁₂₈ (see above)
    27. digital (0, 0, 129)-net over F₁₂₈ (see above)
    28. digital (0, 0, 129)-net over F₁₂₈ (see above)
    29. digital (0, 0, 129)-net over F₁₂₈ (see above)
    30. digital (0, 0, 129)-net over F₁₂₈ (see above)
    31. digital (0, 0, 129)-net over F₁₂₈ (see above)
    32. digital (0, 0, 129)-net over F₁₂₈ (see above)
    33. digital (0, 0, 129)-net over F₁₂₈ (see above)
    34. digital (0, 0, 129)-net over F₁₂₈ (see above)
    35. digital (0, 0, 129)-net over F₁₂₈ (see above)
    36. digital (0, 0, 129)-net over F₁₂₈ (see above)
    37. digital (0, 0, 129)-net over F₁₂₈ (see above)
    38. digital (0, 0, 129)-net over F₁₂₈ (see above)
    39. digital (0, 0, 129)-net over F₁₂₈ (see above)
    40. digital (0, 0, 129)-net over F₁₂₈ (see above)
    41. digital (0, 0, 129)-net over F₁₂₈ (see above)
    42. digital (0, 0, 129)-net over F₁₂₈ (see above)
    43. digital (0, 0, 129)-net over F₁₂₈ (see above)
    44. digital (0, 0, 129)-net over F₁₂₈ (see above)
    45. digital (0, 0, 129)-net over F₁₂₈ (see above)
    46. digital (0, 0, 129)-net over F₁₂₈ (see above)
    47. digital (0, 0, 129)-net over F₁₂₈ (see above)
    48. digital (0, 0, 129)-net over F₁₂₈ (see above)
    49. digital (0, 0, 129)-net over F₁₂₈ (see above)
    50. digital (0, 0, 129)-net over F₁₂₈ (see above)
    51. digital (0, 0, 129)-net over F₁₂₈ (see above)
    52. digital (0, 0, 129)-net over F₁₂₈ (see above)
    53. digital (0, 0, 129)-net over F₁₂₈ (see above)
    54. digital (0, 0, 129)-net over F₁₂₈ (see above)
    55. digital (0, 0, 129)-net over F₁₂₈ (see above)
    56. digital (0, 0, 129)-net over F₁₂₈ (see above)
    57. digital (0, 0, 129)-net over F₁₂₈ (see above)
    58. digital (0, 0, 129)-net over F₁₂₈ (see above)
    59. digital (0, 0, 129)-net over F₁₂₈ (see above)
    60. digital (0, 0, 129)-net over F₁₂₈ (see above)
    61. digital (0, 0, 129)-net over F₁₂₈ (see above)
    62. digital (0, 0, 129)-net over F₁₂₈ (see above)
    63. digital (0, 0, 129)-net over F₁₂₈ (see above)
    64. digital (0, 0, 129)-net over F₁₂₈ (see above)
    65. digital (0, 0, 129)-net over F₁₂₈ (see above)
    66. digital (0, 0, 129)-net over F₁₂₈ (see above)
    67. digital (0, 0, 129)-net over F₁₂₈ (see above)
    68. digital (0, 0, 129)-net over F₁₂₈ (see above)
    69. digital (0, 0, 129)-net over F₁₂₈ (see above)
    70. digital (0, 0, 129)-net over F₁₂₈ (see above)
    71. digital (0, 0, 129)-net over F₁₂₈ (see above)
    72. digital (0, 0, 129)-net over F₁₂₈ (see above)
    73. digital (0, 0, 129)-net over F₁₂₈ (see above)
    74. digital (0, 0, 129)-net over F₁₂₈ (see above)
    75. digital (0, 0, 129)-net over F₁₂₈ (see above)
    76. digital (0, 0, 129)-net over F₁₂₈ (see above)
    77. digital (0, 0, 129)-net over F₁₂₈ (see above)
    78. digital (0, 0, 129)-net over F₁₂₈ (see above)
    79. digital (0, 0, 129)-net over F₁₂₈ (see above)
    80. digital (0, 0, 129)-net over F₁₂₈ (see above)
    81. digital (0, 0, 129)-net over F₁₂₈ (see above)
    82. digital (0, 0, 129)-net over F₁₂₈ (see above)
    83. digital (0, 0, 129)-net over F₁₂₈ (see above)
    84. digital (0, 0, 129)-net over F₁₂₈ (see above)
    85. digital (0, 0, 129)-net over F₁₂₈ (see above)
    86. digital (0, 0, 129)-net over F₁₂₈ (see above)
    87. digital (0, 0, 129)-net over F₁₂₈ (see above)
    88. digital (0, 0, 129)-net over F₁₂₈ (see above)
    89. digital (0, 0, 129)-net over F₁₂₈ (see above)
    90. digital (0, 0, 129)-net over F₁₂₈ (see above)
    91. digital (0, 0, 129)-net over F₁₂₈ (see above)
    92. digital (0, 0, 129)-net over F₁₂₈ (see above)
    93. digital (0, 0, 129)-net over F₁₂₈ (see above)
    94. digital (0, 0, 129)-net over F₁₂₈ (see above)
    95. digital (0, 0, 129)-net over F₁₂₈ (see above)
    96. digital (0, 0, 129)-net over F₁₂₈ (see above)
    97. digital (0, 0, 129)-net over F₁₂₈ (see above)
    98. digital (0, 0, 129)-net over F₁₂₈ (see above)
    99. digital (0, 0, 129)-net over F₁₂₈ (see above)
    100. digital (0, 0, 129)-net over F₁₂₈ (see above)
    101. digital (0, 0, 129)-net over F₁₂₈ (see above)
    102. digital (0, 0, 129)-net over F₁₂₈ (see above)
    103. digital (0, 0, 129)-net over F₁₂₈ (see above)
    104. digital (0, 0, 129)-net over F₁₂₈ (see above)
    105. digital (0, 0, 129)-net over F₁₂₈ (see above)
    106. digital (0, 0, 129)-net over F₁₂₈ (see above)
    107. digital (0, 0, 129)-net over F₁₂₈ (see above)
    108. digital (0, 0, 129)-net over F₁₂₈ (see above)
    109. digital (0, 0, 129)-net over F₁₂₈ (see above)
    110. digital (0, 0, 129)-net over F₁₂₈ (see above)
    111. digital (0, 0, 129)-net over F₁₂₈ (see above)
    112. digital (0, 0, 129)-net over F₁₂₈ (see above)
    113. digital (0, 0, 129)-net over F₁₂₈ (see above)
    114. digital (0, 0, 129)-net over F₁₂₈ (see above)
    115. digital (0, 0, 129)-net over F₁₂₈ (see above)
    116. digital (0, 0, 129)-net over F₁₂₈ (see above)
    117. digital (0, 0, 129)-net over F₁₂₈ (see above)
    118. digital (0, 0, 129)-net over F₁₂₈ (see above)
    119. digital (0, 0, 129)-net over F₁₂₈ (see above)
    120. digital (0, 0, 129)-net over F₁₂₈ (see above)
    121. digital (0, 1, 129)-net over F₁₂₈, using
      - s-reduction based on digital (0, 1, s)-net over F₁₂₈ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    122. digital (0, 1, 129)-net over F₁₂₈ (see above)
    123. digital (0, 1, 129)-net over F₁₂₈ (see above)
    124. digital (0, 1, 129)-net over F₁₂₈ (see above)
    125. digital (0, 2, 129)-net over F₁₂₈, using
      - kÂ =Â 2 construction [i]
    126. digital (0, 2, 129)-net over F₁₂₈ (see above)
    127. digital (0, 4, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
    128. digital (0, 8, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
2. (37, 53, 1048575)-net in base 128, using
  - net defined by OOA [i] based on OOA(128⁵³, 1048575, S₁₂₈, 16, 16), using
    - OA 8-folding and stacking [i] based on OA(128⁵³, 8388600, S₁₂₈, 16), using
      - discarding factors based on OA(128⁵³, large, S₁₂₈, 16), using
        discarding parts of the base [i] based on linear OA(256⁴⁶, large, F₂₅₆, 16) (dual of [large, large−46, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(57, 57+16, large)-Net over F₁₂₈ — Digital

Digital (57, 73, large)-net over F₁₂₈, using

t-expansion [i] based on digital (56, 73, large)-net over F₁₂₈, using
- 4 times m-reduction [i] based on digital (56, 77, large)-net over F₁₂₈, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+16, large)-Net in Base 128 — Upper bound on s

There is no (57, 73, large)-net in base 128, because

14 times m-reduction [i] would yield (57, 59, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (57, 57+16, s)-Nets in Base 128

(57, 57+16, 1048833)-Net over F128 — Constructive and digital

(57, 57+16, 1065087)-Net in Base 128 — Constructive

(57, 57+16, large)-Net over F128 — Digital

(57, 57+16, large)-Net in Base 128 — Upper bound on s

(57, 57+16, 1048833)-Net over F₁₂₈ — Constructive and digital

(57, 57+16, large)-Net over F₁₂₈ — Digital